Imagine you are measuring the voltages at different points in the circuit from the "-" end of the battery. The + end of the battery measures obviously 9V. You know the voltage across the 1K resistor is 3V, and one end of it is at 9V. So what is the voltage at the other end?

I was thinking of doing it that way, keeping the 6kOhm and another resistor parallel to it and use the current division method. However, what should I do about the 1kOhm resistor in the way?

You know the total current flowing through the parallel resistors is 3 mA. So you can forget about the battery and the 1K resistor from now on - they are just providing that 3ma current. Work out how the 3mA is split between the two resistors in parallel and you are done.

When I do these current divider type deals, I find it helpful to label the equivalent resistances as I'm reducing them, and draw an arrow to the circuit showing where it applies. This way when I need to do a current divider, or maybe multiple current dividers, then I already have the "2nd resistor" to plug into the equation.